SciPost Phys. 3, 021 (2017)
A short introduction to topological quantum computation
Ville T. Lahtinen1 and Jiannis K. Pachos2?
1 Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany 2 School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom
Abstract
This review presents an entry-level introduction to topological quantum computation – quantum computing with anyons. We introduce anyons at the system-independent level of anyon models and discuss the key concepts of protected fusion spaces and statisti- cal quantum evolutions for encoding and processing quantum information. Both the encoding and the processing are inherently resilient against errors due to their topolog- ical nature, thus promising to overcome one of the main obstacles for the realisation of quantum computers. We outline the general steps of topological quantum computa- tion, as well as discuss various challenges faced by it. We also review the literature on condensed matter systems where anyons can emerge. Finally, the appearance of anyons and employing them for quantum computation is demonstrated in the context of a sim- ple microscopic model – the topological superconducting nanowire – that describes the low-energy physics of several experimentally relevant settings. This model supports lo- calised Majorana zero modes that are the simplest and the experimentally most tractable types of anyons that are needed to perform topological quantum computation.
Copyright V.T. Lahtinen and J. K. Pachos. Received 12-05-2017 This work is licensed under the Creative Commons Accepted 24-08-2017 Check for Attribution 4.0 International License. Published 09-09-2017 updates Published by the SciPost Foundation. doi:10.21468/SciPostPhys.3.3.021
Contents
1 Introduction2 1.1 Topology, stability and anyons3
2 Topological order and anyons in condensed matter systems6 2.1 Topological states that support anyons7 2.2 Manifestations of anyons in microscopic many-body systems9 2.2.1 Degeneracy and Berry phases9 2.2.2 Topological degeneracy and entanglement entropy 11
3 Anyon models 12 3.1 Fusion channels - Decoherence-free subspaces 13 3.2 Braiding anyons - Statistical quantum evolutions 15 3.3 Example 1: Fibonacci anyons 15 3.4 Example 2: Ising anyons 17
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4 Quantum computation with anyons 19 4.1 Initialization of a topological quantum computer 19 4.2 Quantum gates – Braiding anyons 20 4.3 Measurements – Fusing anyons 21 4.4 Possible error sources 22
5 Topological quantum computation with superconducting nanowires 24 5.1 Majorana zero modes in a superconducting nanowire 25 5.2 The Majorana qubit 27 5.3 Manipulating and reading out the Majorana qubit 29 5.4 Challenges with Majorana-based topological quantum computation 30
6 Outlook 30
References 31
1 Introduction
Topological quantum computation is an approach to storing and manipulating quantum infor- mation that employs exotic quasiparticles, called anyons. Anyons are interesting on their own right in fundamental physics, as they generalise the statistics of the commonly known bosons and fermions. Due to this exotic statistical behaviour, they exhibit non-trivial quantum evo- lutions that are described by topology, i.e. they are abstracted from local geometrical details. When anyons are used to encode and process quantum information, this topological behaviour provides a much desired resilience against control errors and perturbations. To be more pre- cise, the presence of certain kinds of anyons gives rise to a degenerate decoherence-free sub- space, in which the state can only be evolved by moving the anyons adiabatically around each other. While from the first sight anyons appear to be an over-complicated method for per- forming quantum computation, they are profoundly linked to quantum error correction [1], the algorithmic means we have in dealing with errors during quantum computation. In a sense, anyonic quantum computers implement quantum error-correction at the hardware level, thus becoming resilient to control errors and erroneous perturbations. This has augmented topo- logical quantum computation from a niche field of research to a methodology that permeates much of the research efforts in realising fault-tolerant quantum computation. In this review we present a non-technical introduction to anyons and to the framework for performing fault-tolerant quantum computation with them. The emphasis will be on the key properties that define anyons and how they enable protected encoding and processing of quantum information. As anyons can emerge in numerous microscopically distinct systems, we discuss these concepts primarily at a platform-independent level of anyon models and provide an extensive list of references for the interested reader to go deeper. Our aim is to provide a clear and concise introduction to the underlying principles of topological quantum computation without expertise in condensed matter theory. When condensed matter concepts are needed, we introduce them in a heuristic level to give the reader a general understanding without referring to the mathematical details. In doing so, we aim this review to be accessible to anyone with a solid undergraduate understanding of quantum mechanics and the basics of quantum information. While several reviews have already been written on the topic, we aim this review to serve as
2 SciPost Phys. 3, 021 (2017) an accessible starting point. Topological quantum computation with fractional quantum Hall States is reviewed extensively by Nayak et al. [2], while Das Sarma et al. focus on quantum computation with Majorana zero modes [3]. The book by Pachos can be viewed as an extended version of the present review that goes deeper into the condensed matter topics [4], while the book by Wang focuses on the more mathematical aspects and their connections to knot theory [5]. The reader may also find useful the lecture notes by Roy and DiVincenzo [6] and the classic lecture notes by Preskill [7]. This review concerns only topological quantum computation where both the encoding and processing of quantum information is topologically protected. Anyons have applications also to quantum memories and quantum error correction, i.e. when only the encoding is topologically protected. For reviews on topological quantum memories, we refer the interested reader to the reviews by Terhal [8] and Brown et al. [9]. This review is structured as follows. We begin in Section 1.1 by describing at a heuristic level why topology can increase fault-tolerance and why the dimension of space is paramount when looking for systems that support anyons. In Section2 we discuss the different types of topological order and the conditions under which they can support anyons. An extensive list of known systems of anyons is provided and we also outline how the defining properties of anyons manifest themselves in microscopic systems. In Section3 we turn to the system independent discussion of anyon models and describe how a minimal set of data captures all the dynamics associated with a given anyon model. As examples we consider both Fibonacci anyons (what we would like to have for topological quantum computation) and Ising anyons (what we have so far). Section4 is the core of the review where we explicitly discuss how Ising anyons can be used to encode and process quantum information in a topologically protected manner, while in Section5 we illustrate how such quantum computation could be carried out in a specific microscopic system. As an example we employ superconducting nanowire arrays that support Majorana zero modes and that are currently the experimentally most promising direction. We conclude with Section6.
1.1 Topology, stability and anyons In mathematics, topology is the study of the global properties of manifolds that are insensitive to local smooth deformations. The overused, but still illustrative example is the topological equivalence between a donut and a coffee cup. Regardless of the local details that give them rather different everyday practicalities, both are mathematically described by genus one man- ifolds meaning that there is a single hole in both. Small smooth deformations, such as taking a bite on the side of the donut or chipping away a piece of the cup will change the object locally, but the topology remains unchanged. Only global violent deformations, such as cutting the donut in half or breaking the cup handle, will change the topology by removing the hole. However, in real world small deformations matter. Quite spectacular salesmanship is re- quired to sell a donut from which someone has already taken a bite. Something similar occurs also in quantum mechanics. To store and evolve a pure quantum state coherently, one must take exceptional care that no outside noise interferes and that the evolution is precisely the desired one. This is the key fundamental challenge in quantum computation: to robustly store quantum states for long times and evolve them according to specific quantum gates. Were quantum information encoded in topological properties of matter, and were the quan- tum gates dependent only on the topology of the evolutions, then both should be inherently protected from local perturbations. Such topological quantum computation would exhibit in- herent hardware-level stability that ideally would make elaborate schemes of quantum error- correction redundant. This idea was first floated by Kitaev in connection to surface codes for quantum error correction [1,10]. He realized that certain codes could be viewed as spin lattice models, where the elementary excitations are anyons – quasiparticles with statistics interpolating between
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� �
� � 3D 2D
Figure 1: Exchange statistics in 2D vs. 3D. In 3D the path λ2 describing two particle exchanges is continuously deformable to λ1 by taking it behind or front of the right-most particle, and in turn λ1 is contractible to a point. Hence, all the paths have the same topology and thus correspond to the same statistical quantum evolution. In 2D, however, the paths λ2 and λ1 are topologically inequivalent since λ2 can not be deformed through the right-most particle, while λ1 is still contractible to a point. Hence, the paths now have different topology and different statistical quantum evolutions can be assigned to each.
those of bosons and fermions [11]. By manipulating these excitations, quantum states could be encoded in the global properties of the system and manipulated by transporting the anyons along non-contractible paths. The local nature of the paths would be irrelevant – any two paths that were topologically equivalent implemented the same quantum gate. This insight put the study of anyons at the center of topological quantum computation. Importantly, it gave significant renewed incentive to condensed-matter physicists to look for realistic systems that could give rise to them. The reason why anyons can exist in general can be traced back to the simple, but far reach- ing realization that local physics should remain unchanged when two identical particles are exchanged. In three spatial dimensions (3D) this dictates that only bosons and fermions can exist as point-like particles. A wave function describing the system of either types of particles acquires a +1 or a 1 phase, respectively, whenever they are exchanged. However, when one goes down to two− spatial dimensions (2D), a much richer variety of statistical behaviour is allowed. In addition to bosonic and fermionic exchange statistics, arbitrary phase factors, or even non-trivial unitary evolutions, can be obtained when two particles are exchanged. This fundamental difference between 2D and 3D arises due to the different topology of space-time evolutions of point-like particles. Consider the exchange processes of two particles illustrated in Figure1. In 3D the path λ2 drawn by the encircling particle is always continuously de- formable to the path λ1 that does not encircle the other particle (the path can be deformed to pass behind the other particle). This loop, in turn, is fully contractible to a point, which means that the wave function of the system must satisfy
3D : Ψ(λ2) = Ψ(λ1) = Ψ(0) . (1) | 〉 | 〉 | 〉 As one particle encircles the other twice, the evolution of the system can be represented by the 2 exchange operator R such that Ψ(λ2) = R Ψ(0) . The contractibility of the loop requires 2 that R = 1, which has only the| solutions〉 R = | 1 that〉 correspond to the exchange statistics of either bosons or fermions. This means that the± order and the orientation of the exchanges are not relevant and the statistics of point-like particles in 3D are mathematically described by the permutation group. This contrasts with 2D, where the path λ2 is no longer continuously deformable (the path is not allowed to pass through the encircled particle) to the fully contractible path λ1. This means that the final state Ψ(λ2) no longer needs to equal the initial state Ψ(0) | 〉 | 〉 2D : Ψ(λ2) = Ψ(λ1) = Ψ(0) . (2) | 〉 6 | 〉 | 〉
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Φ Φ
q q
Figure 2: Toy model for anyons as charge-flux composites where magnetic flux Φ confined to a tube that is encircled by a ring of electric charge q. When one such composite object moves around the other, its charge (flux) circulates the flux (charge) of the other anyon. The Aharonov-Bohm effect gives rise to the complex phase e2iqΦ, which describes the mutual statis- tics of the composite objects. If 2qΦ is not an integer multiple of 2π, the composite objects are Abelian anyons.
Hence the exchange operator R is no longer constrained to square to identity either. Instead, it can be represented by a complex phase, or even a unitary matrix. In the first case the anyons are called Abelian anyons due to their exchange operators commuting, while in the latter case 1 the anyons are referred to as non-Abelian anyons. Since one no longer demands R = R− , the order and orientation of the exchanges are physical and the only constraints on the exchange operator R are given by consistency conditions for distinct evolutions. These derive from a mathematical structure known as the braid group, which describes all topologically distinct evolutions of point-like particles in two spatial dimensions. It is this description of the 2D statistics by the braid group, instead of the permutation group, that allows anyons to exist. While the possibility of something to exist is not equivalent to it actually existing, this simple analysis gives the key hint for where to look for anyons – systems that are 2D. As we are living in a 3D world, no genuine 2D systems exist. Nevertheless, many systems can be constrained to exhibit effective 2D behavior, such as electron gases at 2D interfaces of 3D ma- terials, isolated sheets of atoms such as graphene or 2D optical lattices of cold atoms. One should keep in mind though that 2D only enables anyons to exist, but by no means guarantees that. In fact, the emergence of anyons requires further special conditions, that can be illus- trated using an intuitive toy model for anyons [12]. Consider a composite particle that consists of a magnetic flux Φ confined inside a small solenoid and a ring of electric charge q around it, as illustrated in Figure2. If one such quasiparticle encircles the other, then its charge q goes around the flux Φ, and vice versa. Due to the celebrated Aharonov-Bohm effect [13], the wave function of the system acquires a phase factor e2iqΦ, even if there is no direct interaction between the quasiparticles. Since all the magnetic flux is confined to the solenoid, this phase factor does not depend in the local details of the path, which makes it topological in nature. Thus the wave function will evolve exactly in the same way as a system of two particles with iqΦ exchange statistics described by R = e . If
qΦ = πn, n = 1, 2, . . . , (3) 6 then the composite particles are Abelian anyons. For instance, if the flux is given by half of a flux quantum Φ = π, then any fractionalized charge q = n (in units of the electron charge) makes the flux-charge composites anyons. While this is6 a toy picture, it hints of an intimate connection between anyons and fractionalisation. Whether fractionalized quasiparticles emerge in a given microscopic system is a model specific question that has no universal answer. Most of the known systems require strong interactions between the elementary particles, such as electrons. However, also defects in
5 SciPost Phys. 3, 021 (2017) certain non-interacting states can give rise to anyons, as is the case with superconducting nanowires that we discuss in Section5. We now turn to discuss different types of 2D topological states of matter and review the literature on those that either support or are proposed to support anyons.
2 Topological order and anyons in condensed matter systems
Nowadays it is impossible to talk of distinct phases of matter without talking about topology. Topological insulators and superconductors, Weyl or Dirac semi-metals, both integer and frac- tional quantum Hall states and spin liquids are all instances of topological states of matter. Without going into the details, mathematically every topological state of matter is character- ized by some topological invariant, that can be calculated from the ground state wavefunction and that takes different values in different states. However, when it comes to supporting anyons, not all topological states are equal. In this subsection we give a brief overview of distinct topological states of matter, the general features required for them to support anyons and list those for which there is either theoretical or experimental evidence. We conclude the section by outlining how the defining properties of anyons manifest themselves in such microscopic systems. Topological states of matter come in two broad distinct classes. The first class consists of states that are topological only given that some protecting symmetry is respected. If the symmetry is broken, the states immediately become trivial. Hence, such states are com- monly referred to as symmetry-protected topological (SPT) states. They include all integer quantum Hall states and topological insulators and superconductors (for reviews we refer to [14, 15]), that have been classified based on the fundamental symmetries of time-reversal and charge-conjugation [16] as well as based on symmetries arising from the underlying crys- tal lattice [17–20]. All these are systems of non-interacting fermions, but SPT states can also emerge in bosonic systems, such as spin chains or lattice models, and in the presence of inter- actions [21–23]. While SPT states exhibit interesting phenomena, such as protected surface currents even if the bulk is an insulator, they do not support anyons as intrinsic quasiparti- cle excitions. However, there is an exception to this rule if the systems are allowed to have defects, such as domain walls between different states of matter, vortices in a superconduc- tor or lattice dislocations. These may bind localized zero energy modes that can behave as anyons. In particular, in topological superconductors localized zero energy modes are de- scribed by Majorana (real) fermions that can be viewed as fractionalized halves of complex fermion modes [27, 28, 54]. For all practical purposes Majorana zero modes behave like non- Abelian anyons, but they are also the most complex kind of anyonic quasiparticles that can emerge in SPT states of free fermions. Unfortunately, they are not universal for quantum com- putation by purely topological means, but as the most experimentally accessible anyons, they have been the subject of intense research efforts. We review briefly these developments below and in Section5 employ them to illustrate how topological quantum computation could be carried out in an array of topological superconducting nanowires. The second class consists of states with intrinsic topological order that does not require any symmetries to be present. This class includes the strongly interacting fractional quantum Hall states and spin liquids. These states always support different kinds of anyons as intrinsic quasi- particle excitations (no defects are required and anyons beyond Majorana zero modes are in principle available), some of which are universal for quantum computation by purely topolog- ical means. Unlike SPT states, ground states of intrinsically topologically ordered states also exhibit long-range entanglement that gives rise to topological entanglement entropy [24–26] and ground state degeneracy that depends on the topology of the manifold the system is de-
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fined on [29]. While these concepts are not directly related to topological quantum compu- tation, they are of paramount importance in identifying the presence of intrinsic topological order in different models and we review them briefly in Section 2.2.
2.1 Topological states that support anyons As discussed above, anyons require either states with intrinsic topological order or SPT states with some kind of defects. Regarding the latter, of particular interest are topological super- conductors that support Majorana zero modes. To briefly review the literature on condensed matter systems that support anyons of interest to topological quantum computation, we discuss them under three broad classes: (i) Strongly correlated electron gases under strong magnetic fields that support fractional quantum Hall states, (ii) collective states of strongly interacting spins giving rise to spin liquid states and (iii) topological superconductors and their engineered variants in superconductor-topological insulator / semi-conductor heterostructures.
(i) Fractional quantum Hall states Fractional quantum Hall (FQH) states occur when very cold electron gases are subjected to high magnetic fields. In such states the electrons localize and form so called Landau levels, which makes the system a highly degenerate insulator. Since the electron motion is frozen out, interactions between the electrons become significant. Due to the interactions the gapped ground states at different magnetic fields can be described by a non-integer filling fraction ν – the number of electrons per flux quantum, i.e. fractionalization occurs. While the bulk of the 2D system is insulating, experimentally such states are most easily detected by measuring the Hall conductance as the function of applied magnetic field. This exhibits characteristic plateaus corresponding to different filling fractions [30,31]. Every plateau is a distinct phase of intrinsic topological matter, characterized by the quantized Hall conductance that is proportional to the topological invariant characterizing the state [32]. The nature of these intrinsically topologically ordered states and the anyons they support is understood via trial wave functions. Such wave functions were first proposed by Laughlin to predict that the ν = 1/3 state supports fractionalized quasiparticles that behave as Abelian anyons [33]. While the direct probing of the anyons via their exchange statistics remains elu- sive, the charge fractionalization has been experimentally confirmed, thus strongly supporting the existence of anyons [34, 35]. Numerous further trial wave functions have been proposed to describe other filling fractions seen in the experiments. From the point of view of quantum computation, of particular interest is the ν = 5/2 case. It has been proposed that at this filling fraction the system is described by the Moore-Read state that supports the simplest non-Abelian anyons – the Ising anyons [36]. The predicted charge fractionalization has been confirmed, but again direct probing of the anyons has remained elusive [38]. For most practical purposes Ising anyons are equivalent to Majorana modes and hence they are not universal for quantum computation. For universality one needs more complex Fibonacci anyons. These are expected to emerge in the very fragile ν = 12/5 filling fraction described by the Read-Rezayi state [37]. For a comprehensive account of topological quantum computation in FQH states, we refer the interested reader to [2]. It has also been proposed that FQH states can emerge in topological insulators when they are subjected to similar conditions, i.e. strong magnetic fields, strong interactions and frac- tional filling [39]. While such fractional topological (Chern) insulators are an intriguing al- ternative, it is currently unclear whether these conditions can ever be achieved in crystalline materials.
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(ii) Spin liquids When strong Coulombic interactions localize electrons in a lattice configuration, their kinetic degrees of freedom are frozen out in a Mott insulating state. However, the electron spins still interact and can form collective states that exhibit intrinsic topological order. Such states are known as topological spin liquids [40]. The study of these systems roughly follows two distinct routes. The bottom-up route is to study common spin-spin interactions, which usually are of Heisenberg type, on distinct 2D lattices. As such systems rarely lend themselves to analytic treatment, one employs mean-field theory to understand what phases could exist [41], and re- lies on state-of-the-art numerics to verify the predictions. Convincing numerical evidence has been obtained that topological spin liquids that support Abelian anyons do exist on several frustrated 2D lattices (e.g. triangular or Kagome lattices) [42–46]. The top-down route is to write down idealized spin lattice models that support a given topological phase. The canon- ical model of this type is Kitaev’s honeycomb model [47] that supports Ising anyons akin to the Moore-Read state. The model is exactly solvable, which enables the emergent anyons to be studied in detail, but the fine-tuning required for the exact solvability also means that it is unlikely to appear in nature. However, certain compounds have been proposed to be de- scribed by a perturbed version of the model [48,49], and neutron-scattering experiments have provided initial evidence for spin liquid states in these materials [50]. Generalizations of the honeycomb model exist both in 2D and 3D [51,52]. As these systems all follow the same construction, this family of models is still the only analytically tractable framework for spin liquids. In principle, spin liquids supporting any types of anyons can be defined on lattices via the quantum-double [10] or the Levin-Wen [53] construction. How- ever, these require replacing actual spins with more generic local degrees of freedom subject to rather unphysical constraints and many-body interactions. A celebrated example of the quantum-double construction is the so called Toric Code [10]. While being an intrinsically topologically ordered states with Abelian anyons, the Toric Code is also the simplest example of a topological quantum memory that features heavily in relation to quantum memories and error correction (for reviews we refer to [8,9 ]).
(iii) Topological superconductors in heterostructures
Since the seminal work by Read and Green [54], it has been known that if time-reversal sym- metry is broken and the pairing in a 2D superconductor is so-called p-wave type, then vortices (the natural defects in superconductors) bind Majorana zero modes. While actual materials exhibiting such pairing are yet to be found (though strontium ruthenate is strongly believed to be one) it was realized that qualitatively same physics could occur when a topological in- sulator [55], a spin-orbit coupled semiconductor [56,57], a chain of magnetic atoms [58–60] or half-metals [61,62] is placed in the proximity of a regular s-wave superconductor. In other words, the combination of physics from both systems realizes effective p-wave superconduc- tivity at the interface. Following an early proposal by Kitaev [63], wires made of these materials, when deposited on top of a superconductor, were predicted to host Majorana modes at their ends, which could be probed through simple conductance measurements [64,65]. While the explicit verification of their braiding properties is yet to be carried out, several experiments on microscopically distinct setups strongly support the existence of Majorana modes [66–71]. These topological nanowire heterostructures are the most prominent candidate to experimentally test the key building blocks of topological protection and implementation of topological gates via the ex- change statistics of anyons. For a comprehensive review of Majorana zero modes in solid-state systems, see e.g. [72–75].
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(iv) Quantum simulations of anyonic systems In addition to looking for anyons in materials, much progress has been made in simulating topological states of matter with cold atoms in optical lattices [76]. Time-reversal symmetry broken Chern insulators [77,78] have been realized and it is hoped that these systems can be pushed towards fractionalized conditions that support anyons. Proposals also exist for Kitaev’s honeycomb model [79], as well as for counterparts of topological superconducting nanowires that host Majorana zero modes [80–82]. The statistical properties of Majorana zero modes can also be simulated in cavity arrays [83] and photonic quantum simulators [84, 85]. These systems are only unitarily equivalent to actual topological states of matter and hence not genuinely topological in nature. However, they still realize counterparts of the protected subspaces and statistical evolutions in the pres- ence of anyons. This makes them attractive for experimentally testing the required control to reliably manipulate quantum information in a topological-like encoding.
2.2 Manifestations of anyons in microscopic many-body systems We close this section by discussing how non-Abelian anyons manifest themselves in intrinsically topologically ordered microscopic systems and highlight the similarities / differences to SPT states with defects. The common key property is the emergence of a protected degenerate subspace, where the evolution is given as a non-Abelian Berry phase when the anyons are adiabatically moved around each other. The precise structure of this protected space and the possible evolutions depend on the types of anyons and they are discussed in Section3. We also introduce two often appearing concepts – topological entanglement entropy and topological ground state degeneracy – that are not directly relevant to topological quantum computation, but which are important diagnostic tools in identifying the presence of intrinsic topological order.
2.2.1 Degeneracy and Berry phases All topological states that support anyons are insulators. By this we mean that the ground state is separated from the rest of the states in the spectrum by a spectral gap ∆. When a topo- logical system is placed on a surface of trivial topology without boundaries (we discuss the possible degeneracy that can arise from non-trivial spatial topologies in the following subsec- tion), such as a sphere, then the ground state is unique. However, when non-Abelian anyonic quasiparticles are introduced into the system, the lowest energy state in their presence exhibits degeneracy that depends on the types of anyons, as illustrated in Figure3. This contrasts with Abelian anyons, in whose presence the lowest energy state remains unique. This non-local degenerate manifold of states is in general exponentially degenerate in the anyon separation (degeneracy lifting anyon-anyon interactions are discussed in Section 4.4) and it is separated from any other states by the spectral gap ∆. This degenerate subspace is a collective non-local property of the non-Abelian anyons that one employs to encode quantum information in a topologically protected manner. This pro- tection arises from the presence of an energy gap and from the non-locality. Since all the excitations in the system are massive (in SPT states anyons are massless zero modes, but the defects on which they are bound to are still massive), the energy gap suppresses spontaneous excitations in the system that could interfere with the already present anyons and thereby change the state in the degenerate subspace. The non-locality protects the state in the de- generate manifold via the realistic assumption that any noise in the system acts locally and hence may only result in local displacement of the anyons. As long as this displacement is small compared to their separation, no evolution takes place in the protected subspace. Thus
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E
Excited states Excited states Δ Δ
E0 + 4mσ Unique ground state Degenerate low-energy manifold E0
σ σ σ σ
Figure 3: When an intrinsically topologically ordered state is placed on a surface of trivial topology and no anyons are present, the ground state at some energy E0 is unique and sepa- rated from excited states by an energy gap ∆. In the presence of non-Abelian anyons that we denote by σ, the lowest energy state exhibits degeneracy that depends on the type of anyons present (e.g. four-fold degeneracy if four σ anyons of the Ising model are present, as we dis- cuss soon in Section 3.4). Since anyons are massive excitations in intrinsically topologically ordered states, this degenerate manifold appears at some higher energy than the ground state in the absence of anyons (e.g. at energy E0 +4mσ when each Ising anyon has mass mσ), but it is still separated from excited states by the spectral gap ∆. Qualitatively similar picture applies also to SPT states with defects, such as a topological p-wave superconductor with vortices. In this case each σ denotes a massless Majorana modes bound to a vortex, but each vortex is still carries some mass mσ. the degenerate low-energy manifolds in the presence of non-Abelian anyons are essentially decoherence-free subspaces. States in the protected subspace evolve only when the anyons are transported around each other. Due to the presence of the energy gap, when this process is performed adiabatically, i.e. slowly compared to ∆, the system only evolves in the non-local subspace and is given by to the statistics of the anyons. Microscopically, such evolution due to an adiabatic change in the system is given as a non-Abelian Berry phase acquired by the wave function [127–129]. Let us consider a system of N non-Abelian anyons that give rise to a D-dimensional protected subspace. This space is spanned by D degenerate many-body states given by
Ψn(z1, z2,..., zN ) , n = 1, 2, . . . , D, (4) | 〉 that depend in general on the anyon coordinates zj. Let λ be a cyclic path in zj that winds one anyon around another, as illustrated in Figure4. If one changes the parameters zj slowly in time compared to the energy gap ∆, then the transport is adiabatic and the system evolves only within the degenerate ground state manifold spanned by the states (4). This evolution is in general given by
D X Ψn(z1, z2,..., zN ) Γnm(λ) Ψm(z1, z2,..., zN ) , (5) | 〉 → m=1 | 〉 where the non-Abelian Berry phase is defined by I Γ (λ) = P exp A dz. (6) λ ·
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λ σ σ σ σ z1 z2 z3 z4
�(�) ≈ � � �
Figure 4: Microscopic braiding as a non-Abelian Berry phase. Consider a system where two pairs of anyons, each denoted by a σ, are created from the vacuum (the pairs are connected by dashed lines) located at positions z1,..., z4. Transporting the anyon at position z2 along any path λ that encloses only the anyon at z3 gives rise to non-Abelian Berry phase (6) that acts in the degenerate subspace shown in Figure3. When the anyons correspond to Ising anyons, to be discussed in detail in Section 3.4, and the transport is adiabatic, this non-Abelian Berry 1 2 phase will accurately approximate their braid matrix F − R F given by (22).
Here P denotes path ordering and the components of the non-Abelian Berry connection are given by j ∂ (A )mn = Ψm(z1, z2,..., zN ) Ψn(z1, z2,..., zN ) . (7) 〈 | ∂ zj | 〉
The geometric phase due to the cyclic evolution in the coordinates zj does not depend on the time it takes to traverse the path λ as long as it is long enough for the evolution to be adiabatic. Nor does it depend on the exact shape of the path λ and thus the unitary Γ (λ) is topological in nature. The precise form it takes depends on the types of anyons present and what their mutual anyonic statistics are. For each anyon model, there is only a finite number of possible unitary evolutions, generated by pairwise exchanges, that act in the protected non-local subspace. We discuss the different types of statistics in Section3. It has been verified in several microscopically distinct settings that the statistics of the anyons is indeed obtained from the adiabatic evolution of their wave functions. Analytically this has been shown for the Laughlin [130] and Moore-Read [131] fractional quantum Hall states, as well as p-wave superconductors [132] including heterostructure realizations of topo- logical nanowires [133, 134]. These calculations are supported by numerics that have been used to demonstrate non-Abelian statistics in more complex fractional quantum Hall states [106,135,136] and in microscopic models such as Kitaev’s honeycomb lattice model [109,137] or bosonic fractional quantum Hall systems [138].
2.2.2 Topological degeneracy and entanglement entropy In Section2 we discussed the two distinct notions of topological order – SPT order and intrinsic topological order. Regarding anyons, the key difference was that the first required some kinds of defects to support, e.g., Majorana zero modes, while the latter does not. There are also two other important physical consequences of intrinsic topological order that are absent in SPT states. The first property is topological ground state degeneracy, i.e. that the ground state in the absence of anyons is degenerate when the system is defined on a manifold of non-trivial spatial topology [29]. In other words, a state defined on a sphere (the spatial manifold has no holes) would have a unique ground state, but the same state defined on a torus (the spatial manifold has a single hole) would have a degenerate ground state space. This is a general property that applies to any intrinsic topological state regardless of the type of anyons, Abelian or non- Abelian, it supports. However, the degree of degeneracy does depend on the anyon model
11 SciPost Phys. 3, 021 (2017) and the topology of the spatial manifold and needs to be worked out on a case-by-case basis [139]. The most important consequence to quantum computation is that, given that non- trivial spatial topologies can be engineered, the degenerate ground state space can be used as a topological quantum memory. In particular, since the nature of the supported anyons is irrelevant, systems supporting relatively simpler Abelian anyons can also be employed to this end. For thorough discussion about topological quantum memories, we refer to [8,9 ]. The second important application is to use the degeneracy on manifolds of distinct topology as a convenient numerical probe to identify the presence of intrinsic topological order [42–46]. The second property inherent to only intrinsic topological states is that of topological entanglement entropy. Any quantum state can be partitioned into two disjoint regions A and B. The entanglement between the regions can be quantified by entanglement entropy S = TrBρ log ρ, where ρ is the density matrix of the full system, but the trace is taken only over− the region B. All gapped states of matter are expected to have only short-range entan- glement and thus follow the area-law scaling of entanglement entropy [140]. In other words, the entanglement between A and B should be proportional to the length ∂ A of the boundary between them. However, if a state exhibits intrinsic topological order,| then| there is also a universal constant correction to the area-law [24–26], with the entanglement entropy scaling as
S = α ∂ A γ, (8) | | − where α is a non-universal constant. On the other hand, γ is a universal constant that takes a non-zero value only in the presence of intrinsic topological order. Like the ground state degeneracy on manifolds of varying topology, γ depends on the types of anyons supported. Extracting it by studying the entropy scaling (8) is thus another convenient numerical tool to identify the presence of intrinsic topological order. However, neither the topological ground state degeneracy nor the topological entanglement entropy are unique characteristics of anyon models – different anyon models can give rise to the same degeneracy and to the same correction to entanglement entropy. To unambiguously determine the anyon model supported by a given state, the statistics of the supported anyons can be obtained from the ground state via more sophisticated manipulations of the state [141–143].
3 Anyon models
From now on we adopt the perspective that anyons exist and focus on explaining what different types of anyons there can be and what their defining properties are. Furthermore, for the time being, we assume that the microscopic details of the system that give rise to the anyons can be completely neglected and the low-energy dynamics is described in terms of only the anyons. Under these assumptions the possible evolutions are limited to three simple scenarios:
1. Anyons can be created or annihilated in pairwise fashion.
2. Anyons can be fused to form other types of anyons.
3. Anyons can be exchanged adiabatically.
The formal framework capturing these properties in a unified fashion goes under the name of a topological quantum field theory [86]. From the point of view of topological quantum computa- tion, most of the details of this rigorous mathematical description can be omitted. As a result, only a minimal set of data is required to specify the properties of the anyons corresponding to a particular topological quantum field theory. Here, as also often is the case in literature,
12 SciPost Phys. 3, 021 (2017) we refer to this minimal information as an anyon model. Such models with up to four distinct anyonic quasiparticles have been systematically classified [87]. We refer the interested reader also to the appendices of [47] for an accessible introduction to the diagrammatic notation often used when talking about anyons. In this section we first give the general structure of anyon models to explain how a given anyon model constrains the nature of the protected subspace discussed in Section 2.2 and the possible evolutions therein. Then we illustrate these abstract concepts with two examples most relevant to quantum computation: Fibonacci anyons (universal for quantum computation, but up to now only a theoretical construction) and Ising anyons (not universal for quantum computation, but can be experimentally realized).
3.1 Fusion channels - Decoherence-free subspaces To define an anyon model, one first specifies how many distinct anyons there are. For com- pleteness, this list must include a trivial label, 1, corresponding to the vacuum with no anyons. The anyon model is then spanned by some number of particles
M = 1, a, b, c,... , (9) { } where the labels a,b,c,. . . can be viewed as topological charges carried by each anyon. As charges they must satisfy conservation rules. For anyons these are known as fusion rules, that take the form X c a b = Nabc, (10) × c M c ∈ where the fusion coefficients Nab = 0, 1, . . . are non-negative integers describing the possible topological charges (fusion channels) a composite particle of a and b can carry (a and b c are fused). In general Nab can be any non-negative integer, but for most physical models c c Nab = 0, 1. If Nab = 0, then fusing a with b can not yield c. If for all a, b M there is only one c ∈ Nab that is different from zero, then the fusion outcome of each pair of particles is unique and the model is Abelian. On the other hand, if for some pair of anyons a and b there are two or c more fusion coefficients that satisfy Nab = 0, then the model is called non-Abelian. The latter implies that the fusion of a and b can result6 in several different anyons, i.e. there is degree of freedom associated with the fusion channel. Furthermore, to conserve total topological 1 charge every particle a must have an anti-particle b, in the sense that Nab = 1 for some b. For instance, a fusion rule of the form a a = 1 + b, that we encounter below, means that a is a non-Abelian anyon, as it has two possible× fusion outcomes, and that it is its own anti-particle, as one of the possible fusion outcomes is the vacuum. The key characteristic of non-Abelian anyons is that the fusion channel degrees of freedom imply a space of states spanned by different possible fusion outcomes. That is, if a and b can fuse to several c M, we can define orthonormal states ab; c that satisfy ∈ | 〉 ab; c ab; d = δcd . (11) 〈 | 〉 If there are N distinct fusion channels in the presence of a pair of particles, the system ex- hibits N-fold degeneracy spanned by these states. We refer to this non-local space shared by the non-Abelian anyons, regardless of where they are located, as the fusion space. This fusion space is precisely the protected low-energy subspace discussed in Section 2.2. Under the as- sumption that all microscopics of the system giving rise to the anyons are decoupled from the low-energy physics, the states in the fusion space are perfectly degenerate. As it is a collective non-local property of the anyons, no local perturbation can lift the degeneracy and it is hence a decoherence-free subspace. As such it is an ideal place to non-locally encode quantum in- formation. We stress that the fusion space arises from the distinct ways anyons can be fused
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a b c a b c e = � f d d
Figure 5: Fusion diagrams and F-matrices. A basis in the fusion space is given by choosing an order in which the anyons are to be fused without exchanging their positions (this results in a unitary evolution in the fusion space as discussed in the next subsection). In the case of three anyons a, b and c that are constrained to fuse to d, there are only two options: Either they are fused pairwise from left to right ( (ab)c; ec; d ) or from right to left ( a(bc); a f ; d ). These | d 〉 | 〉 two bases are related by the unitary matrix Fabc according to (12). The state in one basis is in general a superposition of the basis states in the other basis.
over how they are fused. If two anyons are actually fused and the outcome of the fusion is detected, this would correspond to performing a projective measurement in the fusion space. The fusion space of a pair of non-Abelian anyons can not be used directly to encode a qubit though. The reason is that two states ab; c and ab; d belong to different global topological charge sectors (given by c and d, respectively)| 〉 and| hence〉 can not be superposed. Instead, one needs more than two anyons in the system, such that they can be fused in various different ways that still give the same fusion outcome when all of them are fused. The basis in such higher dimensional fusion space is given by a fusion diagram of a fixed fusion order spanned by all possible fusion outcomes. Choosing a different fusion order is equivalent to a change of basis. Like the familiar Hadamard gate that relates the Z- and X -bases of a qubit, for every anyon model there exists a set of matrices that relate a state in one basis to states in other bases. These so called F-matrices form part of the data of an anyon model and they are obtained as the solutions to a set of consistency conditions known as the pentagon equations [47]. We do not concern ourselves here with the explicit form of the pentagon equations, as their role is to classify different possible anyon models consistent with given fusion rules (10). For known anyon models this data can be found, e.g., in [87]. To illustrate how the F-matrices give structure to the fusion space, consider a case where three anyons a, b and c are constrained always to fuse to d and assume that several interme- diate fusion outcomes are compatible with this constraint. Then there are several fusion states belonging to the same topological charge sector d that can be superposed. For three anyons there are two possible fusion diagrams corresponding to distinct bases. Either one fuses first a and b to give e, in which case the basis states are labeled by e and denoted by (ab)c; ec; d , or one fuses first b and c to give f , in which case the corresponding basis states are| a(bc); a f 〉; d . d | 〉 These two choices of basis must be related by a unitary matrix Fabc as
X d (ab)c; ec; d = (Fabc)e f a(bc); a f ; d , (12) | 〉 f | 〉
d d where (Fabc)e f are the matrix elements of Fabc, and f is summed over all the anyons that b f and c can fuse to, i.e. for which Nbc = 0. The states and the action of the F-matrices are most conveniently expressed in terms6 of fusion diagrams, as illustrated in Figure5. Such diagrams also vividly capture the topological nature of such processes – two diagrams that can be continuously deformed into each other (i.e. no cutting or crossing of the world lines) correspond to the same state of the system.
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3.2 Braiding anyons - Statistical quantum evolutions The fusion space is the collective non-local property of the anyons. To evolve a quantum state in this space, the statistical properties of anyons are employed. When anyons are exchanged, or braided (as the process is often called due to their worldlines forming braids), the state in the fusion space undergoes a unitary evolution. Transporting the anyons along paths that do not enclose other anyons has no effect. For Abelian anyons the fusion space is one dimensional and the only possible evolution is given by a complex phase factor. This phase depends on the type of anyons and whether they are exchanged clockwise or anti-clockwise, but not on the order of the anyon exchanges. In other words, the exchange operator describing exchanging anyons a iθab and b in a clockwise fashion is given by Rab = e for some statistical angle 0 θab < 2π. This contrasts with non-Abelian anyons for which the resulting statistical phase not≥ only depends on the types of anyons, but also on their fusion channel c, i.e. the exchange is described by the c c iθab operator Rab = e . Thus, when there is fusion space degeneracy associated with different fusion channels, braiding assigns different phases to different fusion channels and depends not only on the orientation of the exchanges, but also on their order. Given the F-matrices of the anyon model, the possible statistics described by the exchange c operators Rab compatible with them can be obtained by solving another set of consistency equations known as hexagon equations [47, 87]. These R-matrices, as they are often called, describe all possible unitary evolutions that can take place in the fusion space. Consider again the case where anyons a, b and c are constrained to fuse to d, as in (12). Then in the basis (ab)c; ec; d a clockwise exchange of a and b implements the unitary | 〉 X f (ba)c; ec; d = Rabδe,f (ab)c; ec; d , (13) | 〉 f | 〉 where f spans all possible fusion outcomes of a and b and δe,f is the Kronecker delta function. Thus for a generic state in this basis, a clockwise exchange is represented by a diagonal unitary f matrix R with entries Rab. Were the anyons exchanged counter-clockwise, the evolution would be described by R†. To write down the effect of exchanging b and c clockwise in the same basis, d one first applies the Fabc to change the basis to a(bc); a f ; d , then applies R, as defined above, d| 1 〉 and then returns to the original basis with (Fabc)− , which is given by the unitary evolution
d 1 d (ac)b; ec; d = (Fabc)− R(Fabc) (ab)c; ec; d . (14) | 〉 | 〉 All unitary evolutions due to distinct exchanges of anyons can be constructed in similar fash- ion and they are always given by some combination of the F- and R-matrices. Again, these exchange operations are conveniently expressed diagrammatically, as illustrated in Figure6. Summarizing, an anyon model is most compactly specified by: (i) The fusion coefficients c Nab that describe how many distinct anyons there are and how the anyons fuse, (ii) the F- matrices that describe the structure of the fusion space and (iii) the R-matrices that describe the mutual statistics of the anyons. Regardless of the microscopics that give rise to anyons in a given system, with this minimal set of data all possible states of the fusion space for arbitrary number of anyons and all the possible evolutions can be constructed. As this notation gets quickly rather cumbersome, we illustrate it with two examples that are most relevant to topological quantum computation.
3.3 Example 1: Fibonacci anyons Based on their anyon model structure, Fibonacci anyons are the simplest non-Abelian anyons. There is only one anyon τ that satisfies the fusion rule
τ τ = 1 + τ. (15) × 15 SciPost Phys. 3, 021 (2017)
b a c a b c e = � e
d d
a c b a b c e = � � (� ) e
d d
Figure 6: Exchanging anyons in different bases and the R-matrices. When two anyons a and b are exchanged in a basis where they are fused first ( ab c; ec; d ), the R-matrix acts as a e ( ) e iθab | 〉 diagonal matrix that assigns a phase factor Rab = e depending on their fusion channel e. When b and c are exchanged, the action in the basis where a and b are fused first is obtained d by first moving to the basis they are used first ( a(bc); a f ; d ) by applying the unitary Fabc, then applying the diagonal R-matrix that assigns| different phase〉 factors to different fusion d 1 channels of b and c and finally returning to the original basis by applying (Fabc)− .
In other words, τ is its own anti-particle, but two τ anyons can also behave like a single τ anyon. Repeated associative application of the fusion rule shows that the dimension of the fusion space, i.e. the number of different ways the fusion of all τ anyons can result in either total topological charge of 1 and τ, grows in a rather peculiar manner
τ τ τ = 1 + 2 τ, × × · τ τ τ τ = 2 1 + 3 τ, (16) × × × · · τ τ τ τ τ = 3 1 + 5 τ, × × × × · · and so on. In other words the dimensionality of the fusion space in both sectors grows as the Fibonacci series, where the next number is always the sum of the two preceeding numbers (hence the name Fibonacci anyons!). This immediately points to a peculiarity of the Fibonacci fusion space. It does not have a natural tensor product structure in the sense of its dimension- ality growing by a constant factor per added τ anyon. To encode a qubit in the fusion space of Fibonacci anyons, we see from (16) that one needs three τ anyons that are constrained to fuse to a single τ particle (or equivalently, four τ anyons constrained to the total vacuum sector). A basis in this two-dimensional fusion subspace is given by the states (ττ)τ; 1τ; τ and (ττ)τ; ττ; τ , with the corresponding fusion diagrams given by Figure5 with| the corresponding〉 | label substitutions.〉 For the Fibonacci fusion rules the F-matrix giving the basis transformation and the R-matrix describing braiding are given by
4πi 1 1/2 1 5 τ φ− φ− Rττ 0 e 0 F = Fτττ = 1/2 1 , R = τ = 3πi , (17) 0 R − 5 φ− φ− ττ 0 e − where φ = (1+p5)/2 is the Golden Ratio (another characteristic of the Fibonacci series). For a detailed discussion on Fibonacci anyons, we refer to [88]. This seeming simplicity of the Fibonacci anyons hides something remarkable though. Namely, arbitrary unitaries can be implemented by braiding the Fibonacci anyons and hence 10 the model is universal for quantum computation [2]! Even if R equals the identity matrix,
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1 the braid group generated by R and F − RF is dense in SU(2) in the sense that an arbitrary uni- taries can be approximated to arbitrary accuracy by only braiding the anyons. There are two serious caveats though. First, the lack of tensor product structure means that only a subspace of the full fusion space can be used to encode information. For instance, if three τ anyons are used to encode a qubit, then one would like to use altogether six anyons to encode two qubits. However, their fusion space is five-dimensional in the vacuum sector, which means that the logical qubits reside only in a subspace. The second caveat is that approximating even the simplest gates is far from straighforward. Even the NOT-gate requires thousands of braiding operations in very specific order [89, 90]. Several techniques exist to construct the required braids more efficiently [91–95], but the task remains challenging. Still, the fact that braiding can be employed to generate in principle arbitrary unitaries, as opposed to most other anyon models, makes Fibonacci anyons the Holy Grail for quantum computation. While the caveats can be overcome, the most daunting thing about Fibonacci anyons is that their relative simplicityas an anyon model by no means correlates with the accessability of the microscopic systems that support them. Quite the contrary. While novel elaborate schemes to realize them have been proposed in coupled domain wall arrays of Abelian FQH states [96], the most plausible candidate is still the Read-Rezayi state that has been proposed to describe the filling fraction ν = 12/5 FQH state [37]. As this state is very fragile, it remains unclear whether it can ever be realized in a laboratory. Hence, much research has focused on states hosting simpler non-Abelian anyons that might not be universal, but which still enable testing and development of topological quantum technologies. In this regard Ising anyons are of particular interest.
3.4 Example 2: Ising anyons The Ising anyon model consists of two non-trivial particles ψ (fermion) and σ (anyon) satis- fying the fusion rules 1 1 = 1, 1 ψ = ψ, 1 σ = σ, × ψ ψ =×1, ψ σ =×σ, × σ σ = 1 +×ψ. × The fusion rule ψ ψ = 1 implies that, when brought together, two fermions behave like there is no particle, while× ψ σ = σ implies that ψ with a σ is indistinguishable from a single σ. The non-Abelian nature× of the σ anyons is encoded in the last fusion rule, which says that two of them can behave either as the vacuum or as a fermion. Physically, the fusion rules can be understood, for instance, in the context of a topological p-wave superconductor [54]. There, the vacuum 1 is a condensate of Cooper pairs. The fermions ψ are Bogoliubov quasiparticles that can pair into a Cooper pair and thus vanish into the vacuum. The σ anyons, on the other hand, correspond Majorana zero modes bound to vortices. As we will explain in Section5, a Majorana mode corresponds to a “half” of a complex fermion mode. A pair of such vortices carries thus a single non-local fermion mode, the ψ particle, that can be either unoccupied (fusion channel σ σ 1) or occupied (fusion channel σ σ ψ). The non-Abelian× fusion→ rule for the σ anyons implies that× there→ is a two dimensional fusion space associated with a pair of them. The basis can be associated with the two fusion channels and denoted by σσ; 1 , σσ; ψ . However, as these two states belong to different topo- logical charge sectors,{| they〉 | can not〉} be superposed. In order to have a non-trivial fusion space in the same charge sector, one needs to consider at least three σ particles that can fuse to a single σ in two distinct ways (or equivalently four σ anyons that fuse to 1), as shown by the
17 SciPost Phys. 3, 021 (2017) repeated associative application of the fusion rules
σ σ σ = 2 σ, × × · σ σ σ σ = 2 1 + 2 ψ, (18) × × × · · σ σ σ σ σ = 4 σ, × × × × · and so on. The basis in the constrained fusion space can be given by the states
(σσ)σ; 1σ; σ , (σσ)σ; ψσ; σ , (19) {| 〉 | 〉} that correspond to the two left-most σ anyons fusing into either 1 or ψ. The F-matrix to change the basis to fusing from right to left is given by
1 1 1 F F σ . (20) = σσσ = 1 1 p2 − Since it creates equal weight superpositions, it means that if the fusion outcome is unique in one basis, in the other basis it is completely random. Thus the different fusion orders of Ising anyons correspond to different bases exactly as the basis for a qubit could be chosen along the Z-axis or along X -axis. Unlike Fibonacci anyons, the fusion space of Ising anyons has a natural tensor product structure. We see from (18) that the dimension of the fusion space doubles for every added σ pair and hence for 2N anyons the dimension of the fusion space in N 1 a fixed topological charge sector is given by 2 − . The R-matrix describing the statistics of Ising anyons under the clockwise exchange of the two left-most σ anyons is given by
1 Rσσ 0 i π 1 0 R ψ e 8 π . (21) = = − i 2 0 Rσσ 0 e
We immediately see that if we encoded a qubit in the fusion space associated with three σ particles, R2 would implement the logical phase-gate up to an overall phase factor. If the two right-most anyons were instead exchanged twice, the evolution in the basis (19) would be described by 0 1 F 1R2 F e iπ4 . (22) − = − 1 0 In other words, braiding them changes the fusion channel of the two left most ones between 1 and ψ. If the three σ’s were employed to encode a qubit, this braid would have implemented a logical NOT-gate. The limitation of the operations that can be performed is obvious from the braiding of three σ anyons. Since one can only implement logical phase and NOT-gates on a single qubit, Ising anyons can only implement the Clifford group by braiding [97, 98]. This means that Ising anyons, while being non-Abelian, are not universal for quantum computation by braid- ing. To overcome this shortcoming, non-topological schemes have been devised to promote their computational power to universality. While the need for such non-topological operations makes the system more susceptible to errors, Ising anyons are still the best candidates to ex- perimentally test the principles of topological quantum computation due to their realization as Majorana zero modes in superconducting nanowire heterostructures [66–70]. In the Section 4 we describe in more detail how a topological quantum computer would in general be oper- ated. In Section5 illustrate these steps in the context of the experimentally relevant Majorana wires.
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4 Quantum computation with anyons
While discussing non-Abelian anyons and the non-local fusion space associated with them, we have already hinted how this space could be used to encode and process quantum information in an inherently fault-resilient manner. In the ideal conditions of zero temperature and infinite anyon separation, the states in the fusion space have three very attractive properties:
(i) All the states are perfectly degenerate.
(ii) They are indistinguishable by local operations.
(iii) They can be coherently evolved by braiding anyons.
If this space of states is used as the logical space of a quantum computer, property (i) implies that the encoded information is free of dynamical dephasing, while property (ii) means that it is also protected against any local perturbations. Property (iii) means that errors could only occur under unlikely non-local perturbations to the Hamiltonian that would create virtual anyons and propagate them around the encoding ones. However, braiding of the anyons can still be carried out robustly by the operator of the computer to execute desired quantum gates. Furthermore, property (iii) implies that all the quantum gates are virtually free from control errors since they depend only on the topological characteristics of the braiding evolutions given by the F- and R-matrices. Together these properties mean that quantum computation with anyons would heavily suppress errors already at the level of the hardware, with little need for resource intensive quantum error correction. Of course, in the real world these ideal conditions are never met and some decoherence of the encoded information always takes place. Still, the topological encoding and processing of quantum information provides in principle unparalleled protection over non-topological schemes. For the time being we forget about the nasty reality and focus on outlining the steps to operate a topological quantum computer with Ising anyons as our case study. Generic error sources present in a laboratory are discussed at the end of the section.
4.1 Initialization of a topological quantum computer To initialize a quantum computer, one needs first to identify the computational space of n qubits. In topological computer this means creating some number of anyons from the vacuum and fixing their positions. The system then exhibits a fusion space manifesting as a protected non-local subspace that is identified as the computational space. Depending on the anyons in question, the full fusion space may not admit a tensor product structure (such as the Fibonacci anyons), but one can always identify subspaces corresponding to the fusion channels of subsets of anyons that can serve as qubits. To illustrate the steps of operating a topological quantum computer, we focus on the simpler Ising anyons whose fusion space does exhibit natural tensor product structure. As discussed in Section 3.4, for 2N Ising anyons the fusion space dimension in a fixed N 1 topological charge sector increases exponentially as D = 2 − . To initialize the system, we assume that σ anyons are created pairwise from the vacuum with no other anyons present. This means that every pair is initialized in the σ σ 1 fusion channel and hence the system globally belongs also to the vacuum sector. Thus four× →σ anyons encodes a qubit, six σ’s encodes two qubits, and so on. Let us focus on a system of six σ anyons that enables to demonstrate all the basic operations. It is convenient to choose the pairwise fusion basis as a computational
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σ σ σ σ σ σ a b c a b c |�� 〉 1 1 1 |�� 〉 � � 1 |�� 〉 1 � � |�� 〉 � 1 � 1
Figure 7: The fusion diagram for six Ising anyons for a pairwise basis restricted to the global vacuum sector. Due to the fusion rules σ σ = 1 + ψ and σ ψ = σ, the fusion diagram contains four distinct fusion channels that are× consistent with the× constraint that the fusion of all the six σ anyons must give the vacuum 1. The table shows the identification of the fusion channels with the computational basis of two qubits. basis
00 = σσ; 1 σσ; 1 σσ; 1 , | 〉 | 〉 | 〉 | 〉 10 = σσ; ψ σσ; ψ σσ; 1 , | 〉 | 〉 | 〉 | 〉 01 = σσ; 1 σσ; ψ σσ; ψ , (23) | 〉 | 〉 | 〉 | 〉 11 = σσ; ψ σσ; 1 σσ; ψ , | 〉 | 〉 | 〉 | 〉 where the three kets refer to the fusion channels of the three σ pairs, as illustrated in Figure 7. When the anyon pairs are created from the vacuum, the topological quantum computer is initialized in the state 00 . The other basis states involve an even number of intermediate ψ channels, which according| 〉 to the fusion rule ψ ψ = 1 is consistent with the constraint that the fusion of all σ particles must always yield the× vacuum.
4.2 Quantum gates – Braiding anyons To perform a computation in the fusion space is equivalent to specifying a braid – a sequence of exchanges of the anyons that corresponds to the desired sequence of logical gates. For Ising anyons, the natural gate set to implement consists of Clifford operations on single qubits, i.e. the X -, Z- and Hadamard UH -gates and the two-qubit controlled phase gate UCZ . The single qubit gates follow directly from the F- and R-matrices of Ising anyons, (21) and (22). The first says that under two exchanges the state acquires a 1 phase whenever the exchanged σ pair fuses to a ψ, while the latter says that exchanging− twice two σ anyons from different pairs simultaneously changes the fusion channel of both pairs between 1 and ψ. In the two-qubit computational basis (23), the elementary single qubit operations, up to an overall phase, are thus given by 2 1 2 2 2 X1 = R23 = F − R F I, Z1 = R12 = R I, 2 1 ⊗2 2 ⊗ 2 X2 = R45 = I F − R F, Z2 = R56 = I R , (24) ⊗ ⊗ where Ri j is the clockwise exchange operator acting on anyons i and j. The corresponding braiding diagrams are illustrated in Figure8. Similarly, the Hadamard gates UH can be imple- 1 mented by single exchanges. One can easily verify that F − RF creates superpositions of fusion channels, but with distinct phase factors assigned to different fusion channels. These phase factors can be cancelled by appending it further exchanges. A little algebra shows that up to an overall phase, the Hadamard gates on the two qubits are given by 1 UH,1 = R12R23R12 = RF − RFR I, 1 ⊗ UH,2 = R56R45R56 = I RF − RFR, (25) ⊗
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